Cantorjoukko

Cantorjoukko, also known as the Cantor set, is a fascinating mathematical object in set theory and fractal geometry. It is a set of points on the real number line that exhibits self-similarity and has several counterintuitive properties.

The construction of the Cantor set begins with the closed interval [0, 1]. In the first step, the open middle third of this interval, (1/3, 2/3), is removed. This leaves two closed intervals: [0, 1/3] and [2/3, 1]. In the second step, the open middle third of each of these remaining intervals is removed. This means removing (1/9, 2/9) and (7/9, 8/9). This process is repeated infinitely. At each step, the open middle third of every existing closed interval is removed.

Despite being constructed by repeatedly removing parts of the original interval, the Cantor set is not empty. It contains an uncountable infinity of points. Furthermore, the total length of the removed intervals sums to 1, meaning the Cantor set has a Lebesgue measure of zero. This means it occupies "no space" in terms of length, yet it is infinitely complex.

The Cantor set is also totally disconnected, meaning that no two distinct points in the set can be connected by a path that stays entirely within the set. Each point in the Cantor set can be uniquely represented by a ternary (base-3) expansion consisting only of digits 0 and 2. This property highlights its structure and its connection to other mathematical concepts.